Building Root Data
- sub<R | a>: RootDtm, SetEnum -> RootDtm
The root subdatum of the root datum \(R\) generated by the roots \(\alpha_{a_1},\dots,\alpha_{a_k}\) where \(a=\{a_1,\dots,a_k\}\) is a set of integers.
- sub<R | s>: RootDtm, SetEnum -> RootDtm
The root subdatum of the root datum \(R\) generated by the roots \(\alpha_{s_1},\dots,\alpha_{s_k}\) where \(s=[s_1,\dots,s_k]\) is a sequence of integers. In this version the roots must be simple in the root subdatum (i.e. none of them may be a summand of another) otherwise an error is signalled. The simple roots will appear in the subdatum in the given order.
- Example: Root Subdata (ex-dc9697)
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> R := RootDatum("A4"); > PositiveRoots(R); {@ (1 0 0 0), (0 1 0 0), (0 0 1 0), (0 0 0 1), (1 1 0 0), (0 1 1 0), (0 0 1 1), (1 1 1 0), (0 1 1 1), (1 1 1 1) @} > s := sub< R | [6,1,4] >; > s; Root datum of type A3 > PositiveRoots(s); {@ (0 1 1 0), (1 0 0 0), (0 0 0 1), (1 1 1 0), (0 1 1 1), (1 1 1 1) @} > s := sub< R | [1,5] >; Error: The given roots are not simple in a subdatum > s := sub< R | {1,5} >; > s; Root datum of type A2 > PositiveRoots(s); {@ (1 0 0 0), (0 1 0 0), (1 1 0 0) @}
- R1 subset R2: RootDtm, RootDtm -> BoolElt, .
Returns
trueif and only if the root datum \(R_1\) is a subset of the root datum \(R_2\). If true, returns an injection as sequence of roots as second return value.
- R1 + R2: RootDtm, RootDtm -> RootDtm
- DirectSum(R1, R2): RootDtm, RootDtm -> RootDtm
The external direct sum of the root data \(R_1\) and \(R_2\). The full (co)root space of the result is the direct sum of the full (co)root spaces of \(R_1\) and \(R_2\).
- R1 join R2: RootDtm, RootDtm -> RootDtm
The internal direct sum of the root data \(R_1\) and \(R_2\). The root data must have the same full (co)root space, which will also be the full (co)root space of the result. The root data must have disjoint (co)root spaces.
- Example: Sums of Root Data (ex-67182b)
> R := RootDatum("A1A1"); > R1 := sub<R|[1]>; > R2 := sub<R|[2]>; > R1 + R2; Root datum of dimension 4 of type A1 A1 > R1 join R2; R: Adjoint root datum of dimension 2 of type A1 A1
- DirectSumDecomposition(R): RootDtm -> [], RootDtm, Map
- IndecomposableSummands(R): RootDtm -> [], RootDtm, Map
Returns a sequence \(Q\) of irreducible root data, a root datum \(S\) which is the direct sum of the terms of \(Q\), and an isogeny map \(\phi:S\to R\). The root datum \(R\) must be semisimple. Note that a semisimple root datum \(R\) need not be a direct sum of simple root data, but it is isogenous to a direct sum of root data \(S\).
- Example: Decomposing a Root Datum (ex-7e44ad)
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If the root datum in adjoint or simply connected, then it is a direct sum of simples. In this case we get \(S=R\).
> R := RootDatum("A4B5" : Isogeny:="SC"); > Q, S := DirectSumDecomposition( R ); > R eq S; true > R eq Q[1] join Q[2]; true
The join of the summands of the direct sum decomposition is the original root datum again:
> R eq &join DirectSumDecomposition(R); true > R eq &+ DirectSumDecomposition(R); false
> R1 := RootDatum("A3T2B4T3"); > R2 := RootDatum("T3G2T4BC3"); > R1 + R2; Adjoint root datum of dimension 24 of type A3 B4 G2 BC3 > R1 join R2; Root datum of dimension 12 of type A3 B4 G2 BC3
Here is an example of a semisimple root datum which is not a direct sum of simple subdata. Note that a simple root datum of type \(A_1\) is either simply connected or adjoint.
> G<a,b>:=FundamentalGroup("A1A1"); > _,inj:=sub<G|a*b>; > R:=RootDatum("A1A1":Isogeny:=inj); > ad := RootDatum( "A1" : Isogeny:="Ad" ); > sc := RootDatum( "A1" : Isogeny:="SC" ); > IsIsomorphic( R, DirectSum(ad,ad) ); false > IsIsomorphic( R, DirectSum(ad,sc) ); false > IsIsomorphic( R, DirectSum(sc,sc) ); false > Q, S := DirectSumDecomposition( R ); > R eq S; false
- Dual(R): RootDtm -> RootDtm, Map
The dual of the root datum \(R\), obtained by swapping the roots and coroots. The second value returned is the dual morphism from \(R\) to its dual.
- SimplyConnectedVersion(R): RootDtm -> RootDtm, Map
The simply connected version of the root datum \(R\). If \(R\) is semisimple then the injection of the simply connected version into \(R\) is returned as the second value.
- AdjointVersion(R): RootDtm -> RootDtm, Map
The adjoint version of the root datum \(R\). If \(R\) is semisimple then the projection from \(R\) to its adjoint version is returned as the second value.
- IndivisibleSubdatum(R): RootDtm -> RootDtm
The root datum consisting of all indivisible roots of the root datum \(R\).
- Radical(R): RootDtm -> RootDtm
The radical of the root datum \(R\), ie, the toral subdatum whose root (resp. coroot) space consists of the vectors perpendicular to every coroot (resp. root).
- Example: Direct Sums, Duals, and Radicals (ex-187258)
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An adjoint or simply connected root datum is always a direct sum of irreducible subdata. In these cases we take \(S=R\).
> R1 := RootDatum("A5"); > R2 := RootDatum("B4"); > R := DirectSum(R1, Dual(R2)); > DirectSumDecomposition(R); { Root datum of type A5 , Root datum of type C4 }
> R := RootDatum("BC2"); > I := IndivisibleSubdatum(R); I; I: Root datum of type B2 > I subset R; true [ 1, 2, 3, 5, 7, 8, 9, 11 ]
> R := StandardRootDatum("A", 3); > Radical(R); Toral root datum of dimension 1
- TwistedRootDatum(R): RootDtm -> RootDtm
- TwistedRootDatum(N): MonStgElt -> RootDtm
Twist: Any Default: 1
Create a twisted root datum from the root datum \(R\), or from the semisimple root datum with Cartan name \(N\). The twist may be specified in any of the following ways:
An integer, specifying the order of the twist;
A permutation, specifying the action of the primitive roots;
A pair \(<D, i>\), where \(D\) is a set of distinguished orbits as sets of integers, and \(i\) is the order of the Dynkin diagram symmetry;
A pair \(<\Gamma, Q>\), where \(\Gamma\) is the acting group, and \(Q\) is a sequence containing the permutation of the primitive roots for each of the generators of \(\Gamma\);
A homomorphism from \(\Gamma\) to the symmetric group whose order is the number of roots of \(R\), describing how the acting group \(\Gamma\) acts on the roots.
- Example: Constructing Twisted Root Data (ex-4dc407)
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We construct a twisted root datum in a number of ways.
> S := TwistedRootDatum("D4" : Twist := 3); > S; S: Twisted adjoint root datum of dimension 4 of type 3D4,2 > R := RootDatum("A1A3"); > DynkinDiagram(R); A1 1 A3 2 - 3 - 4 > S := TwistedRootDatum(R : Twist := Sym(4)!(2,4)); > S; S: Twisted adjoint root datum of dimension 4 of type 2(A1 A3)4,3 > S := TwistedRootDatum("A4" : Twist := <{{1,4},{2,3}}, 2>); > S; S: Twisted adjoint root datum of dimension 4 of type 2A4,2 > R := RootDatum("E6" : Isogeny := "SC"); > DynkinDiagram(R); E6 1 - 3 - 4 - 5 - 6 | 2 > S := TwistedRootDatum(R : Twist := <Sym(2) ,[ Sym(6)!(1,6)(3,5) ]>); > S; S: Twisted simply connected root datum of dimension 6 of type 2E6,4 > R := RootDatum("D4"); > DynkinDiagram(R); D4 3 / 1 - 2 \\ 4 > Gamma := Sym(3); > Gamma.1, Gamma.2; (1, 2, 3) (1, 2) > S := TwistedRootDatum(R : Twist := <Gamma, [ Sym(4) | (1,3,4), (1,4) ]>); > S; S: Twisted adjoint root datum of dimension 4 of type 6D4,2 > R := RootDatum("A2"); > DynkinDiagram(R); A2 1 - 2 > Roots(R); {@ (1 0), (0 1), (1 1), (-1 0), ( 0 -1), (-1 -1) @} > S6 := Sym(#Roots(R)); > phi := hom<Sym(2) -> S6 | S6!(1,2)(4,5)>; > S := TwistedRootDatum(R : Twist := phi);